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Recurrent Reachability Analysis in Regular Model Checking

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Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2008)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5330))

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Abstract

We consider the problem of recurrent reachability over infinite systems given by regular relations on words and trees, i.e, whether a given regular set of states can be reached infinitely often from a given initial state in the given transition system. Under the condition that the transitive closure of the transition relation is regular, we show that the problem is decidable, and the set of all initial states satisfying the property is regular. Moreover, our algorithm constructs an automaton for this set in polynomial time, assuming that a transducer of the transitive closure can be computed in poly-time. We then demonstrate that transition systems generated by pushdown systems, regular ground tree rewrite systems, and the well-known process algebra PA satisfy our condition and transducers for their transitive closures can be computed in poly-time. Our result also implies that model checking EF-logic extended by recurrent reachability predicate (EGF) over such systems is decidable.

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References

  1. Abdulla, P.A., Jonsson, B.: Undecidable verification problems for programs with unreliable channels. Inf. Comput. 130(1), 71–90 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abdulla, P.A., Jonsson, B., Mahata, P., d’Orso, J.: Regular tree model checking. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 555–568. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  3. Abdulla, P.A., Jonsson, B., Nilsson, M., Saksena, M.: A survey of regular model checking. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 35–48. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  4. Baeten, J., Weijland, W.: Process Algebra. In: CUP (1990)

    Google Scholar 

  5. Benedikt, M., Libkin, L., Neven, F.: Logical definability and query languages over ranked and unranked trees. ACM Trans. Comput. Logic 8(2), 11 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blumensath, A., Grädel, E.: Automatic structures. In: LICS 2000, pp. 51–60 (2000)

    Google Scholar 

  7. Boigelot, B., Legay, A., Wolper, P.: Iterating transducers in the large. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 223–235. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  8. Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of pushdown automata: Application to model-checking. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  9. Bouajjani, A., Legay, A., Wolper, P.: Handling liveness properties in (ω-)regular model checking. In: INFINITY, pp. 101–115 (2004)

    Google Scholar 

  10. Caucal, D.: On the regular structure of prefix rewriting. In: Arnold, A. (ed.) CAAP 1990. LNCS, vol. 431, pp. 61–86. Springer, Heidelberg (1990)

    Google Scholar 

  11. Caucal, D.: On the regular structure of prefix rewriting. Theor. Comput. Sci. 106(1), 61–86 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Comon, H., et al.: Tree Automata: Techniques and Applications (2007), http://www.grappa.univ-lille3.fr/tata

  13. Dauchet, M., Tison, S.: The theory of ground rewrite systems is decidable. In: LICS 1990, pp. 242–248 (1990)

    Google Scholar 

  14. Esparza, J., Hansel, D., Rossmanith, P., Schwoon, S.: Efficient algorithms for model checking pushdown systems. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 232–247. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  15. Esparza, J., Kucera, A., Schwoon, S.: Model checking LTL with regular valuations for pushdown systems. Inf. Comput. 186(2), 355–376 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Esparza, J., Podelski, A.: Efficient algorithms for pre* and post* on interprocedural parallel flow graphs. In: POPL 2000, USA, pp. 1–11. ACM, New York (2000)

    Google Scholar 

  17. Jonsson, B., Nilsson, M.: Transitive closures of regular relations for verifying infinite-state systems. In: Schwartzbach, M.I., Graf, S. (eds.) TACAS 2000. LNCS, vol. 1785, pp. 220–234. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  18. Löding, C.: Model-checking infinite systems generated by ground tree rewriting. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002. LNCS, vol. 2303, pp. 280–294. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  19. Löding, C.: Reachability problems on regular ground tree rewriting graphs. Theory Comput. Syst. 39(2), 347–383 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lugiez, D., Schnoebelen, P.: The regular viewpoint on PA-processes. Theor. Comput. Sci. 274(1-2), 89–115 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lugiez, D., Schnoebelen, P.: Decidable first-order transition logics for PA-processes. Inf. Comput. 203(1), 75–113 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mayr, R.: Process rewrite systems. Inf. Comput. 156(1-2), 264–286 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mayr, R.: Decidability of model checking with the temporal logic EF. Theor. Comp. Sci. 256(1-2), 31–62 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Thomas, W.: Constructing infinite graphs with a decidable MSO-theory. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 113–124. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  25. Vardi, M.Y.: An automata-theoretic approach to linear temporal logic. In: Proc. Banff Higher-Order Workshop, pp. 238–266

    Google Scholar 

  26. Vardi, M.Y., Wolper, P.: Automata-theoretic techniques for modal logics of programs. JCSS 32(2), 183–221 (1986)

    MathSciNet  MATH  Google Scholar 

  27. Walukiewicz, I.: Model checking CTL properties of pushdown systems. In: Kapoor, S., Prasad, S. (eds.) FST TCS 2000. LNCS, vol. 1974, pp. 127–138. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. LFCS, School of Informatics, University of Edinburgh, UK

    Anthony Widjaja To & Leonid Libkin

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  1. Anthony Widjaja To
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  2. Leonid Libkin
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Editors and Affiliations

  1. Carnegie Mellon University, Doha, Qatar

    Iliano Cervesato

  2. Technische Universität Darmstadt, Germany

    Helmut Veith

  3. University of Manchester, UK

    Andrei Voronkov

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To, A.W., Libkin, L. (2008). Recurrent Reachability Analysis in Regular Model Checking. In: Cervesato, I., Veith, H., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2008. Lecture Notes in Computer Science(), vol 5330. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89439-1_15

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  • DOI: https://doi.org/10.1007/978-3-540-89439-1_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-89438-4

  • Online ISBN: 978-3-540-89439-1

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